Stages of Learning Geometry sequential each level has its own language and set of symbols what is implicit of a class of shapes at one level is explicit at the next level e g students use the properties of a class of shapes when classifying at Level 0 but they only begin to isolate and describe them when thinking at Level 1 and levels are subject to reduction by substituting a rote procedure for thinking Level Description Example 0 Visual Judges shapes by their appearances 1 Analysis Sees figures in terms of their components and discovers properties of a class of shapes 2 Informal Deduction Logically interrelates previously discovered properties 3 Deduction Proves theorems deductively 4 Rigor Establishes theorems in different postulational systems Students need to move from one level to the next within each of the topics van Hieles proposed a sequence of five phases of learning These provide teachers a structure for organizing classroom instruction in geometry as described below Level Description Example 0 Inquiry Students discuss and develop questions on a topic to be explored 1 Directed Orientation Students explore sets of carefully sequenced activities 2 Explicitation Students express explicit views and questions about inherent structures of their investigations 3 Free

Manipulatives Evaluation Standards Second Grade Book Addenda Series Grades K 6 pg 25 Manipulatives should be incorporated into activities to help students explore concepts and build understanding Manipulatives can be specifically designed and purchased such as Polydrons or as simple as a piece of paper Manipulatives needed for TEAMS lessons are listed at the beginning of each student program Use of manipulatives is not a guarantee that students will be able to make connections with the mathematics taught It is up to teachers to facilitate lessons to help students discover and apply the mathematical concepts In order to learn mathematics children have to actually do mathematics for themselves rather than learn to follow how someone else does it Beyond Arithmetic Changing Mathematics in the Elementary Classroom pg 13 Suggestions for Implementation Give students time to explore a manipulative in an unstructured manner This gives students a chance to get the play out of their system and to prepare them for using the manipultive as a tool Take time to handle the manipulatives you Have rules for use and care of manipulatives Non compliance could be exclusion from using manipulatives for a short period of time Begin with a specific manipulative for

Cooperative Learning higher achievement for students improved self concept a positive attitude toward school greater retention of material presented more time on task improved higher level thinking improved social skills working with others Hints for Implementing Cooperative Learning Set up rules and expectations for groups to use Start with pairs and then work up to larger groups Select groups to suit the activity Groups can be selected randomly for short term work or heterogeniously for extended group work Arrange the room to facilitate close seating Assign each person in the group a job cheerleader materials specialist timekeeper etc Change the groups after they have been successful at their task Assess each student as well as the collective work of the group Working together is a skill that students need to learn The table below lists levels of behaviors to expect as your student work in cooperative groups Social Skill Behaviors Beginning Basic Advanced Stays with group Checking Criticize ideas not people Helps complete the group s work Shares feelings Non evaluative language descriptive language Listens to others Asks for hlep Perspective taking Shares materials Encourages others Conflict resolution Shares ideas Works well with all Problem solving Takes turns Communication Resolve controversy Uses

Communication mathematics curriculum should include opportunities to communicate so that students can model situations using oral written concrete pictorial graphical and algebraic methods reflect on and clarify their own thinking about mathematical ideas and situations develop common understandings of mathematical ideas including the role of definitions use the skills of reading listening and viewing to interpret and evaluate mathematical ideas discuss mathematical ideas and make conjectures and convincing arguments appreciate the value of mathematical notation and its role in the development of mathematical ideas The ability to read write listen think creatively and communicate about problems will develop and deepen students understandings of mathematics This standard highlights the need to involve children in actively doing mathematics Exploring investigating describing and explaining mathematical ideas promote communication NCTM Curriculum and Evaluation Standards for School Mathematics Reston VA The Council 1989 p 26 Questioning Techniques What questions we ask students and what questions we stimulate students to ask are important for a deeper understanding of mathematical concepts We want students to explore think discover and reason and it is up to the teacher to stimulate and facilitate this The following is an excerpt from Professional Standards For Teaching Mathematics National Council of Teachers of Mathematics March 1991 pp 3 4 It is a good guide to help teachers develop their questioning techniques One would expect to see teachers asking and stimulating students to ask questions like the following Helping students work together to make sense of mathematics What do others think about what Janine said Do you agree Disagree Does anyone have the same answer but a different way to explain it Would you ask the rest of the class that question Do you understand what they are saying Can you convince the rest of us that that makes sense Helping students to rely

Writing in the Mathematics Classroom The journal topic is presented by the studio teacher The classroom teacher can modify the writing assignment but writing should be done at the end of every program Learning logs are also used in TEAMS mathematics programs These logs encourage students to list vocabulary and questions they have related to the activity then to record their understanding as they explore and investigate concepts Why Write Writing as a part of

Technology in the Mathematics Program end of the project a final product is shared back on the web Activities Activities provide online educational experiences to support and extend concepts and skills Activities are designed to involve students in thinking skills how to approach a problem or situation and how to analyze it Activities also involve students in both critical thinking and logical reasoning Math At Home Suggestions and Web Resources for parents Also Activities relating to the TEAMS math lessons being taught in the classroom that parents and their children can do at home Discussion Groups Also known as forums and newsgroups These groups are online discussions on specific topics The discussions are ongoing and replies can be posted at any time The purpose of TEAMS discussion groups is to exchange views about important issues share information and or ask questions Web Links Collections of links to Web pages around the world related to program topics are provided to give TEAMS students access to valuable information that furthers their understanding Student Work Students and teachers are able to send in selected student products and have them posted on the Web These products are scanned and posted on our Shared Work Bulletin Board in the electronic classrooms so they can be seen and enjoyed by everyone Products may include drawings pictures graphs investigation results and HyperStudio stacks In addition links to your school s home page can be created if it features TEAMS student work Photo Album Behind the Scenes Scrapbook The Photo Album is a photo area of the electronic classrooms featuring photos such as students completing a collaborative TEAMS assignment or a visit to your classroom by one of TEAMS Distance Learning Instructors Send your photos or digital images to TEAMS Distance Learning Attention Gayle Perry Please don t send by FAX because of the poor quality of scanned images The Behind the Scene Scrapbook is a photo area featuring photos of the DLIs and production staff preparing for telecasts Teacher Corner The Teacher Corner features useful information on assessment program schedules and description software used on the programs key terms processing of lessons things to try and more One Computer Classroom Many of the software included in the software bibliography have programs that can be done as an entire class activity Spatial sense can be challenged using software as the examples are many Here are some suggestions on how to use the television with one computer to make it happen If you have an Apple II family computer you can simply purchase an audio cable from a store and use the television monitor to display the screen of your computer for the entire class to see The Apple GS has an additional monitor port to use and for the Apple IIe a Y splitter cable with one male and 2 female connections can be purchased to maintain the view on the computer monitor while using the television screen If you have a Macintosh or a PC you will need to purchase a box

Equity in Diverse Population are encouraged to use examples from the variety of cultures and groups represented in their classrooms to illustrate key concepts principles generalizations and theories in their mathematics programs Suggestions for providing equity in diverse populations include arrange for staff development that expands teacher understanding of student differences culture gender learning styles socio economic etc make sure ALL students are given equal time and equal quality instruction use cooperative groups set up heterogeneously show respect and insist on students showing respect What is Multicultural Instruction The basic aim of a multicultural instructional program is to help students see themselves and others as having dignity and worth A multicultural program places emphasis on similarities and differences among individuals and groups Similarities are viewed as those characteristics which make people human and differences are viewed as those characteristics which make each person or group unique and special In this context differences are viewed as positive Thus students are helped to respect and accept a wide range of diversity including physical differences gender differences emotional differences cultural differences and differences in life styles among individuals and groups Student Attitudes 1 The student will indicate feelings of self worth and self acceptance and will demonstrate respect and acceptance of the value dignity and worth of individuals and groups different from himself or herself 2 The student will show acceptance of the validity of cultural pluralism of diverse ways of meeting human needs and of alternative beliefs manners customs and life styles 3 The student will demonstrate that he or she values cultural pluralism as a positive component of the local community the state the nation and the world 4 The student will show evidence of a desire to reduce or eliminate inequalities caused by stereotyping prejudice discrimination and inequality of opportunity 5 The student will